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    "# 初等代数核心计算公式大全（完整版）\n",
    "\n",
    "## 一、基本运算律\n",
    "\n",
    "1. **交换律**\n",
    "   - 加法：$a + b = b + a$\n",
    "   - 乘法：$a \\times b = b \\times a$\n",
    "\n",
    "2. **结合律**\n",
    "   - 加法：$(a + b) + c = a + (b + c)$\n",
    "   - 乘法：$(a \\times b) \\times c = a \\times (b \\times c)$\n",
    "\n",
    "3. **分配律**\n",
    "   - 左分配律：$a \\times (b + c) = a \\times b + a \\times c$\n",
    "   - 右分配律：$(a + b) \\times c = a \\times c + b \\times c$\n",
    "\n",
    "## 二、指数运算规则\n",
    "\n",
    "1. **基本运算**\n",
    "   - $a^m \\times a^n = a^{m+n}$\n",
    "   - $\\frac{a^m}{a^n} = a^{m-n}$ $(a \\neq 0)$\n",
    "   - $(a^m)^n = a^{m \\times n}$\n",
    "\n",
    "2. **特殊规则**\n",
    "   - $a^0 = 1$ $(a \\neq 0)$\n",
    "   - $a^{-n} = \\frac{1}{a^n}$\n",
    "   - $a^{1/n} = \\sqrt[n]{a}$\n",
    "\n",
    "## 三、乘法公式\n",
    "\n",
    "1. **平方公式**\n",
    "   - $(a + b)^2 = a^2 + 2ab + b^2$\n",
    "   - $(a - b)^2 = a^2 - 2ab + b^2$\n",
    "   - **平方差公式**：$a^2 - b^2 = (a + b)(a - b)$\n",
    "\n",
    "2. **立方公式**\n",
    "   - $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$\n",
    "   - $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$\n",
    "   - $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$\n",
    "   - $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$\n",
    "\n",
    "## 四、二次函数相关公式\n",
    "\n",
    "1. **一般式**：$y = ax^2 + bx + c$\n",
    "   - 顶点坐标：$(\\frac{-b}{2a}, \\frac{-b^2}{4a} + c)$\n",
    "   - 对称轴：$x = \\frac{-b}{2a}$\n",
    "2. **顶点式**：$y = a(x - h)^2 + k$\n",
    "   - 顶点坐标：$(h, k)$\n",
    "   - 对称轴：$x = h$\n",
    "3. **求根公式**：\n",
    "   $$x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$$\n",
    "4. **判别式**：$\\Delta = b^2 - 4ac$\n",
    "   - $\\Delta > 0$：两个不等实数根\n",
    "   - $\\Delta = 0$：两个相等实数根\n",
    "   - $\\Delta < 0$：无实数根\n",
    "5. **根与系数关系（韦达定理）**：\n",
    "   - $x_1 + x_2 = -\\frac{b}{a}$\n",
    "   - $x_1 \\times x_2 = \\frac{c}{a}$\n",
    "\n",
    "## 五、分式运算\n",
    "\n",
    "1. **基本运算**\n",
    "   - $\\frac{a}{b} \\pm \\frac{c}{d} = \\frac{ad \\pm bc}{bd}$\n",
    "   - $\\frac{a}{b} \\times \\frac{c}{d} = \\frac{ac}{bd}$\n",
    "   - $\\frac{a}{b} \\div \\frac{c}{d} = \\frac{ad}{bc}$\n",
    "\n",
    "2. **化简规则**\n",
    "   - $\\frac{ab}{ac} = \\frac{b}{c}$ $(a \\neq 0, c \\neq 0)$\n",
    "   - $\\frac{a}{b} = \\frac{a \\times k}{b \\times k}$ $(k \\neq 0)$\n",
    "\n",
    "## 六、根式运算\n",
    "\n",
    "1. **基本规则**\n",
    "   - $\\sqrt{a} \\times \\sqrt{b} = \\sqrt{a \\times b}$\n",
    "   - $\\frac{\\sqrt{a}}{\\sqrt{b}} = \\sqrt{\\frac{a}{b}}$\n",
    "   - $(\\sqrt[n]{a})^m = \\sqrt[n]{a^m}$\n",
    "\n",
    "2. **化简规则**\n",
    "   - $\\sqrt{a^2} = |a|$\n",
    "   - $\\sqrt[n]{a^m} = a^{m/n}$\n",
    "\n",
    "## 七、绝对值性质\n",
    "\n",
    "1. $|a| \\geq 0$\n",
    "2. $|a| = |-a|$\n",
    "3. $|ab| = |a||b|$\n",
    "4. $\\left|\\frac{a}{b}\\right| = \\frac{|a|}{|b|}$ $(b \\neq 0)$\n",
    "\n",
    "## 八、不等式性质\n",
    "\n",
    "1. **传递性**：  \n",
    "   若 $a > b$ 且 $b > c$，则 $a > c$\n",
    "\n",
    "2. **加法保序性**：  \n",
    "   若 $a > b$，则 $a + c > b + c$  \n",
    "   （不等式两边同时加/减同一个数，不等号方向不变）\n",
    "\n",
    "3. **乘法保序性**：\n",
    "   - 若 $a > b$ 且 $c > 0$，则 $ac > bc$  \n",
    "     （不等式两边同乘/除同一个正数，不等号方向不变）\n",
    "   - 若 $a > b$ 且 $c < 0$，则 $ac < bc$  \n",
    "     （不等式两边同乘/除同一个负数，不等号方向**必须改变**）\n",
    "\n",
    "4. **除法保序性**：\n",
    "   - 若 $a > b$ 且 $c > 0$，则 $\\frac{a}{c} > \\frac{b}{c}$\n",
    "   - 若 $a > b$ 且 $c < 0$，则 $\\frac{a}{c} < \\frac{b}{c}$  \n",
    "     （与乘法规则一致，除以负数时不等号方向**必须改变**）\n",
    "\n",
    "5. **倒数性质**：\n",
    "   - 若 $a > b > 0$，则 $\\frac{1}{a} < \\frac{1}{b}$  \n",
    "   - 若 $a < b < 0$，则 $\\frac{1}{a} > \\frac{1}{b}$  \n",
    "   （同号数取倒数时，不等号方向**必须改变**）\n",
    "\n",
    "> **关键记忆点**：  \n",
    "> - 加减法：方向永远不变  \n",
    "> - 乘除法：正数不变方向，负数必须反向  \n",
    "> - 倒数运算：同号数必须反向\n",
    "\n",
    "## 九、比例性质\n",
    "\n",
    "1. 合比定理：若 $\\frac{a}{b} = \\frac{c}{d}$，则 $\\frac{a+b}{b} = \\frac{c+d}{d}$\n",
    "2. 分比定理：若 $\\frac{a}{b} = \\frac{c}{d}$，则 $\\frac{a-b}{b} = \\frac{c-d}{d}$\n",
    "3. 合分比定理：若 $\\frac{a}{b} = \\frac{c}{d}$，则 $\\frac{a+b}{a-b} = \\frac{c+d}{c-d}$\n",
    "\n",
    "## 十、对数运算\n",
    "\n",
    "1. **基本公式**：\n",
    "   - $\\log_a(MN) = \\log_a M + \\log_a N$\n",
    "   - $\\log_a\\left(\\frac{M}{N}\\right) = \\log_a M - \\log_a N$\n",
    "   - $\\log_a M^n = n \\log_a M$\n",
    "\n",
    "2. **换底公式**：\n",
    "   $$\\log_a b = \\frac{\\log_n b}{\\log_n a}$$\n",
    "\n",
    "## 十一、数列求和\n",
    "\n",
    "1. **等差数列**：\n",
    "   - 通项公式：$a_n = a_1 + (n-1)d$\n",
    "   - 前n项和：\n",
    "     $$S_n = \\frac{n(a_1 + a_n)}{2} = \\frac{n[2a_1 + (n-1)d]}{2}$$\n",
    "\n",
    "2. **等比数列**：\n",
    "   - 通项公式：$a_n = a_1 \\times r^{n-1}$\n",
    "   - 前n项和：\n",
    "     $$S_n = \\frac{a_1(1 - r^n)}{1 - r} \\quad (r \\neq 1)$$\n",
    "     $$S_n = n a_1 \\quad (r = 1)$$\n",
    "\n",
    "## 十二、统计与概率\n",
    "\n",
    "### 1. 平均数\n",
    "$$\\bar{x} = \\frac{x_1 + x_2 + \\cdots + x_n}{n}$$\n",
    "\n",
    "### 2. 方差\n",
    "$$s^2 = \\frac{1}{n} \\sum_{i=1}^n (x_i - \\bar{x})^2$$\n",
    "\n",
    "### 3. 概率公式\n",
    "$$P(A) = \\frac{\\text{事件A的可能结果数}}{\\text{所有可能结果数}}$$\n",
    "\n",
    "## 十三、补充公式\n",
    "\n",
    "1. **完全平方公式扩展**：\n",
    "   $$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$$\n",
    "\n",
    "2. **二次函数顶点坐标求法**：\n",
    "   $$h = -\\frac{b}{2a}, \\quad k = c - \\frac{b^2}{4a}$$\n",
    "\n",
    "3. **因式分解技巧**：\n",
    "   - 分组分解法\n",
    "   - 十字相乘法（用于$ax^2 + bx + c$）\n",
    "\n",
    "4. **绝对值不等式**：\n",
    "   - $|x| < a \\Leftrightarrow -a < x < a$\n",
    "   - $|x| > a \\Leftrightarrow x < -a$ 或 $x > a$\n",
    "\n",
    "> 这些公式构成了初等代数的基础，熟练掌握这些公式可以解决绝大多数初等代数问题。建议通过大量练习来加深理解和记忆。"
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